The recently published book Introduction to Galois Theory by David Hernandez and Yves Laszlo is an excellent resource for beginners. "The authors introduce Galois theory from a contemporary point of view. In particular, modern methods such as reduction modulo prime numbers and finite fields are introduced and put to use.

Sep 21, 2017 · And an introduction to fields, and you are ready to start to tackle Galois theory. There are entire books on abstract algebra you can download, and lectures on the subject on You Tube. I have looked at some of the lectures by Benedict Gross on group theory that I liked, but I don't thing he gets to Galois theory. $\endgroup$ –

Nov 2, 2014 · Galois theory is a branch of abstract algebra that gives a connection between field theory and group theory, by reducing field theoretic problems to group theoretic problems. It started out by using permutation groups to give a description of how various roots of a polynomial equation are related, but nowadays, Galois theory has expanded to ...

May 31, 2024 · There is actually an exact analogue of Galois theory in this context, given by the theory of covering spaces in topology. Covering space theory defines a topological version of a (separable) field extension called a covering space , and specialized to graphs, the covering spaces of a graph are always graphs and can be defined purely ...

Read Galois Theory, Rotman, Springer 1998, as a first step. Summing over the comments below, one could conclude that a necessary prerequisite is the collection of such theories as groups, rings, fields, and linear algebra.

Apr 24, 2018 · I'm trying to find applications of Galois theory in different areas. Currently the most interesting for me is the cryptography. As I understood the theory of finite fields (a part of Galois theory) has a strong connection with cryptography. My question is the following:

May 13, 2015 · Can someone give an explanation of what an automorphism is, in the context of Galois Theory? I keep thinking it is the set of maps which send roots of polynomials to their conjugates but I feel that this not the proper explanation of an automorphism. If someone could provide an explanation, and maybe a simple example it would be greatly ...

Feb 10, 2014 · Secondly and more importantly, I have the same problem with this example as with other examples where covering theory or its generalization is treated as a "Galois Theory" (I include the one by Grothendieck here too). This is all fine and well, but any direct connection to the number theory is missing here.

The intuition behind Galois Theory, is that in order to find solutions based on radicals (polynomial functions of the coefficients), one has to do it in a consistent step-by-step manner. In Galois theory one progresively constructs group extensions of a polynomial equation in a step-by-step (or root-by-root) manner.

Aug 5, 2023 · Frobenius elements are a link between Galois theory of number fields and Galois theory of finite fields. When you consider Galois groups of infinite-degree extensions of number fields, not just finite extensions, Galois representations enter the story in a profound way. See, for instance, this MO question and this one too.